How is the band gap really defined and calculated?

Question

When I look into literature I am quite confused on how the band gap of a semiconductor is defined. One statement I often read goes something like that:

The band gap of a semiconductor is the energy difference between the HOMO and LUMO orbital energies.

But does this really make sense? HOMO and LUMO are concepts from molecular orbital theory, describing single electron wavefunctions (they are approximations). However, the total ground state wavefunction (and energy) is not given by a single orbital but by a slater determinant consisting of all occupied orbitals.

My question is:

How is the band gap really defined/calculated? Can Hartree Fock and (non time dependent) DFT be used for this?

My attend: I would suggest that we have to calculate the full wavefunctions of e.g. the ground state and the first excited state. The difference in energy between these two states is then the band gap. The HOMO and LUMO are just used to calculate the full wavefunction. Since Hartree Fock and simple DFT are ground state theories, we cannot calculate the band gap with them - because we do not get excited states.

Edit: I think I have found the solution, but maybe someone can tell me if that is alright: So let’s make the definition that the (fundamental) band gap is defined by: $$E_g = EA - IE$$ Where IE is the ionisation energy and EA is the electron affinity (is this even a valid definition?). The exact way to calculate the band gap is to solve the correlated many electron problem. The solutions are many electron wavefunctions $\phi_i(r_1,r_2…r_n)$ with energy $E_i$. The band gap (or one band gap) is now:$$E_g=E_1-E_0$$ An easier way to solve the problem is with DFT or HF theory (effective single particle theories). DFT is a ground state theory so we do not get excited states. But DFT gives us (by Janak’s theorem) the exact ionisation energy and electron affinity. In DFT we can therefore calculate the band gap (at least the ‚first‘ band gap, there may be more band gaps at higher energies).

In HF theory we can do similar things, but HF theory just gives us the approximate ionisation energy and electron affinity (Koopmans theorem) therefore also the band gap is only approximate.

Some useful things I also found:

• Is Bloch theorem suitable for correlated systems?
• Why is there a band structure for correlated systems?

Answer 1

A band gap is region of a band diagram where no electronic states can exist. Typically, though, a band gap specifically refers to the energy difference between the valence band maximum and conduction band minimum energies.

The quote cited is not generally true. I would guess that they are using a tight-binding model, where the wave functions are are approximated as LCAO’s or molecular orbitals.

Also, the suggestion that the true ground state wave function is a slater determinant is not correct. This is only an approximation in the Hartree-Fock approximation. Hartree-Fock and Kohn-Sham DFT are different approximation schemes for many-body wave functions. In condensed matter physics, Hartree-Fock is rarely used, due to its high computation cost.

Pure Hohenberg–Kohn DFT actually describes a method to exactly solve for the ground state many-body wave function from the electronic density. However, without knowledge of the universal functional, you can’t actually do so. The practical solution to this was the Kohn-Sham formulation of DFT. Though only an approximation of the original Hohenberg–Kohn DFT, with a good approximation of the exchange-correlation functional, one can obtain excellent results.